Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

Cheng, Juan; Shu, Chi‐Wang

doi:10.4208/cicp.030710.131210s

Cited by 24 publications

(22 citation statements)

References 18 publications

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“…We have performed all the tests shown in [13] by the scheme (2.18) and obtained the expected identical results as those in [13]. In this subsection, we will give several new tests which have not been shown in our previous papers to further verify the performance of the scheme and to validate the new implementation.…”

Section: Numerical Testsmentioning

confidence: 69%

“…This discretization can keep the conservation of total energy which has been shown in [13,20]. Thus the proof of conservation is finished.…”

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 81%

“…The above mentioned scheme can keep the spherical symmetry property but can not keep the conservation of momentum or total energy in the cylindrical coordinates. In this paper, we would like to generalize the scheme presented in [13], with the technique described in the previous section, to the LARED-H code which can preserve both the spherical symmetry and conservation properties. In the following, we only discuss the discretization of the terms related to the hydrodynamics, as other terms are solved by the original methods used in the LARED-H code.…”

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 99%

“…As to the property of spherical symmetry, the steps of the proof follow the lines given in [13]. We will not repeat them here to save space.…”

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 99%

“…Thus, we can choose F e as follows, In this subsection, without loss of generality, we choose the cell-centered Lagrangian scheme given in the paper [13], which is a modified version of the cell-centered control volume scheme proposed by Maire in [20], to validate the technique introduced in the previous subsection. This scheme has several good properties such as the conservation of mass, momentum and total energy, compatibility with the geometric conservation law (GCL), robustness and one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid.…”

Section: Introductionmentioning

confidence: 99%

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A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Cheng

Shu

Zeng

2012

Commun. comput. phys.

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Abstract. Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggeredgrid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in * Corresponding author.

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Section: Numerical Testsmentioning

confidence: 69%

“…This discretization can keep the conservation of total energy which has been shown in [13,20]. Thus the proof of conservation is finished.…”

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 81%

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 99%

“…As to the property of spherical symmetry, the steps of the proof follow the lines given in [13]. We will not repeat them here to save space.…”

Section: Three-temperature Model For the Icf Problem And Its Discretimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Cheng

Shu

Zeng

2012

Commun. comput. phys.

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High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

Boscheri

Dumbser

Balsara

2014

Numerical Methods in Fluids

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SUMMARYIn this paper we present a class of high order accurate cell-centered Arbitrary-Eulerian-Lagrangian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes along the principles laid out in [1]. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space-time Galerkin predictor allows the schemes to be high order accurate also in time by using an element-local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained i) either as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu [2], or, ii) as a solution of multiple one-dimensional half-Riemann problems around a vertex, as suggested by Maire [3], or, iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional HLL Riemann solver recently proposed by Balsara et al. [4]. Once the vertex velocity and thus the new node location has been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level t n with the new ones at the next time level t n+1 . If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unneccesary, since the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic MHD equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration.

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An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri

2016

Numerical Methods in Fluids

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Summary In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.

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Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

Cited by 24 publications

References 18 publications

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

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